A cryocontainer is placed in a room at a temperature $T_a.$ The cryocontainer has a wall thickness of $$ and the wall has a thermal conductivity of $k_a.$ The wall has a rectangular shape and the figure below shows the cross$-$sectional view of the wall only. Neglect the end effect. The inner wall surface is $T,$ when the system reaches the steady state. If the thickness of the wall maintains the same while doubling the thermal conductivity of the wall materials from $k_a$ in Case $($a$)$ to $k_b=2k_a$ in Case $($b$).$

$($a$)$ Take the shell balance approach to work out the temperature profile for Case $($a$).$

$($b$)$ Determine the relation between $T_2$ and $T,$ by assuming the heat fluxes in the two cases are the same, i$.$e$.\frac{q_{x,a}}{q_{x,b}}=1.$